https://file.aiquickdraw.com/mathbot/file/1727055372750b0kk6wci.jpg

write x as a linear combination of a and b, round answer to nearest tenth if applicable

To express vector **x** as a linear combination of vectors **a** and **b**, we want to find scalars \( \alpha \) and \( \beta \) such that:

\[
x = \alpha a + \beta b
\]

### From the image:
- The vector **a** points from the origin to approximately (2, -1).
- The vector **b** points from the origin to approximately (2, 2).
- The vector **x** points from the origin to approximately (-7, -5).

Now we need to solve the equation:

\[
\begin{pmatrix} -7 \\ -5 \end{pmatrix} = \alpha \begin{pmatrix} 2 \\ -1 \end{pmatrix} + \beta \begin{pmatrix} 2 \\ 2 \end{pmatrix}
\]

This leads to the system of linear equations:

\[
-7 = 2\alpha + 2\beta
\]
\[
-5 = -\alpha + 2\beta
\]

Let's solve this system to find \( \alpha \) and \( \beta \).The solution to the system of equations is:

\[
\alpha = -\frac{2}{3} \quad \text{and} \quad \beta = -\frac{17}{6}
\]

Rounding these to the nearest tenth, we get:

\[
\alpha \approx -0.7 \quad \text{and} \quad \beta \approx -2.8
\]

Thus, the vector **x** can be written as:

\[
x \approx -0.7a - 2.8b
\]

Would you like more details or have any questions?

Here are some related questions you could explore:
1. What is the geometric meaning of expressing one vector as a linear combination of others?
2. How do we know if vectors **a** and **b** are linearly independent?
3. What happens if **a** and **b** were parallel vectors?
4. How would this method apply in three dimensions?
5. Can we express **x** as a linear combination if **a** and **b** were in different quadrants?

**Tip**: Always check if vectors are linearly independent before expressing one as a combination of others, as this ensures a unique solution.

Write vector x as a linear combination of vectors a and b, rounding the answer to the nearest tenth if applicable.

Learn how to express vector x as a linear combination of vectors a and b. This problem involves solving a system of linear equations to find the coefficients α and β, with vectors a and b from a given image. This tutorial is suitable for advanced high school and college-level students.

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